Question 22

There is one pair (x, y) of integers that satisfies the equation.

The equation \((x - 7)^2 + (y + 3)^2 = 0\) can only hold true when both \((x - 7)^2 = 0\) and \((y + 3)^2 = 0\). This leads to the unique solution where \(x = 7\) and \(y = -3\).

A) None

This option suggests that there are no integer pairs that satisfy the equation. However, the equation can indeed be solved, yielding one specific pair, thus making this option incorrect.

B) One only

This is the correct answer, as solving the equation leads to the unique integer solution \(x = 7\) and \(y = -3\). There is exactly one pair of integers that satisfy the given equation.

C) Two only

This option implies that there are two distinct integer pairs that satisfy the equation. Given that the equation only equals zero at a single point, this option is incorrect.

D) Three only

Indicating that there are three pairs of integers would imply multiple solutions to the equation. Since the equation can only be true for one specific point, this option is also incorrect.

E) More than three

This option suggests an infinite or a greater number of integer pairs satisfying the equation. However, the nature of the equation restricts it to just one solution, making this choice incorrect.

Conclusion

The only solution to the equation \((x - 7)^2 + (y + 3)^2 = 0\) is the integer pair \( (7, -3) \). All other options fail because they suggest the existence of multiple pairs or none at all, which contradicts the derived result of a single valid pair.